Infinitely many small solutions to an elliptic PDE of variable exponent with a singular nonlinearity

We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities \begin{align} (-\Delta)_{p(\cdot)}^{s} u&=\frac{\lambda}{|u|^{\gamma(x)-1}u}+f(x,u)~$\text{in}$~\Omega,\nonumber u&=0~$\text{in}$~$\mathbb{R}$^N\setminus\Omega,\nonumber \end{align} where $\mathrm{\Omega }\subset {\mathbb{R}}^N,N\ge 2$ is a smooth, bounded domain, $\lambda >0$, $s\in (0,1)$, $\gamma (x)\in (0,1)$ for all $x\in \bar{\mathrm{\Omega }}$, $N>sp(x,y)$ for all $(x,y)\in \bar{\mathrm{\Omega }}\times \bar{\mathrm{\Omega }}$ and $(-\mathrm{\Delta }{)}_{p(\cdot )}^s$ is the fractional $p(\cdot )$-Laplacian operator with variable exponent. The nonlinear function $f$ satisfies certain growth conditions. Moreover, we establish a uniform $L^{\mathrm{\infty }}(\bar{\mathrm{\Omega }})$ estimate of the solution(s) by the Moser iteration technique.